3.4 \(\int \frac {e^{\cot ^{-1}(x)}}{(a+a x^2)^3} \, dx\)

Optimal. Leaf size=58 \[ -\frac {(1-4 x) e^{\cot ^{-1}(x)}}{17 a^3 \left (x^2+1\right )^2}-\frac {12 (1-2 x) e^{\cot ^{-1}(x)}}{85 a^3 \left (x^2+1\right )}-\frac {24 e^{\cot ^{-1}(x)}}{85 a^3} \]

[Out]

-24/85*exp(arccot(x))/a^3-1/17*exp(arccot(x))*(1-4*x)/a^3/(x^2+1)^2-12/85*exp(arccot(x))*(1-2*x)/a^3/(x^2+1)

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Rubi [A]  time = 0.07, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5115, 5113} \[ -\frac {(1-4 x) e^{\cot ^{-1}(x)}}{17 a^3 \left (x^2+1\right )^2}-\frac {12 (1-2 x) e^{\cot ^{-1}(x)}}{85 a^3 \left (x^2+1\right )}-\frac {24 e^{\cot ^{-1}(x)}}{85 a^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCot[x]/(a + a*x^2)^3,x]

[Out]

(-24*E^ArcCot[x])/(85*a^3) - (E^ArcCot[x]*(1 - 4*x))/(17*a^3*(1 + x^2)^2) - (12*E^ArcCot[x]*(1 - 2*x))/(85*a^3
*(1 + x^2))

Rule 5113

Int[E^(ArcCot[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> -Simp[E^(n*ArcCot[a*x])/(a*c*n), x] /; Fr
eeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]

Rule 5115

Int[E^(ArcCot[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((n + 2*a*(p + 1)*x)*(c + d*x^
2)^(p + 1)*E^(n*ArcCot[a*x]))/(a*c*(n^2 + 4*(p + 1)^2)), x] + Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 + 4*(p + 1)^2
)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1]
&& NeQ[p, -3/2] && NeQ[n^2 + 4*(p + 1)^2, 0] &&  !(IntegerQ[p] && IntegerQ[(I*n)/2]) &&  !( !IntegerQ[p] && I
ntegerQ[(I*n - 1)/2])

Rubi steps

\begin {align*} \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^3} \, dx &=-\frac {e^{\cot ^{-1}(x)} (1-4 x)}{17 a^3 \left (1+x^2\right )^2}+\frac {12 \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^2} \, dx}{17 a}\\ &=-\frac {e^{\cot ^{-1}(x)} (1-4 x)}{17 a^3 \left (1+x^2\right )^2}-\frac {12 e^{\cot ^{-1}(x)} (1-2 x)}{85 a^3 \left (1+x^2\right )}+\frac {24 \int \frac {e^{\cot ^{-1}(x)}}{a+a x^2} \, dx}{85 a^2}\\ &=-\frac {24 e^{\cot ^{-1}(x)}}{85 a^3}-\frac {e^{\cot ^{-1}(x)} (1-4 x)}{17 a^3 \left (1+x^2\right )^2}-\frac {12 e^{\cot ^{-1}(x)} (1-2 x)}{85 a^3 \left (1+x^2\right )}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 38, normalized size = 0.66 \[ -\frac {\left (24 x^4-24 x^3+60 x^2-44 x+41\right ) e^{\cot ^{-1}(x)}}{85 a^3 \left (x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCot[x]/(a + a*x^2)^3,x]

[Out]

-1/85*(E^ArcCot[x]*(41 - 44*x + 60*x^2 - 24*x^3 + 24*x^4))/(a^3*(1 + x^2)^2)

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fricas [A]  time = 0.63, size = 46, normalized size = 0.79 \[ -\frac {{\left (24 \, x^{4} - 24 \, x^{3} + 60 \, x^{2} - 44 \, x + 41\right )} e^{\operatorname {arccot}\relax (x)}}{85 \, {\left (a^{3} x^{4} + 2 \, a^{3} x^{2} + a^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccot(x))/(a*x^2+a)^3,x, algorithm="fricas")

[Out]

-1/85*(24*x^4 - 24*x^3 + 60*x^2 - 44*x + 41)*e^arccot(x)/(a^3*x^4 + 2*a^3*x^2 + a^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccot(x))/(a*x^2+a)^3,x, algorithm="giac")

[Out]

integrate(e^arccot(x)/(a*x^2 + a)^3, x)

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maple [A]  time = 0.04, size = 36, normalized size = 0.62 \[ -\frac {{\mathrm e}^{\mathrm {arccot}\relax (x )} \left (24 x^{4}-24 x^{3}+60 x^{2}-44 x +41\right )}{85 \left (x^{2}+1\right )^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arccot(x))/(a*x^2+a)^3,x)

[Out]

-1/85*exp(arccot(x))*(24*x^4-24*x^3+60*x^2-44*x+41)/(x^2+1)^2/a^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccot(x))/(a*x^2+a)^3,x, algorithm="maxima")

[Out]

integrate(e^arccot(x)/(a*x^2 + a)^3, x)

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mupad [B]  time = 0.13, size = 53, normalized size = 0.91 \[ -\frac {{\mathrm {e}}^{\mathrm {acot}\relax (x)}\,\left (\frac {41}{85\,a^3}-\frac {44\,x}{85\,a^3}+\frac {12\,x^2}{17\,a^3}-\frac {24\,x^3}{85\,a^3}+\frac {24\,x^4}{85\,a^3}\right )}{x^4+2\,x^2+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(acot(x))/(a + a*x^2)^3,x)

[Out]

-(exp(acot(x))*(41/(85*a^3) - (44*x)/(85*a^3) + (12*x^2)/(17*a^3) - (24*x^3)/(85*a^3) + (24*x^4)/(85*a^3)))/(2
*x^2 + x^4 + 1)

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sympy [B]  time = 6.93, size = 155, normalized size = 2.67 \[ - \frac {24 x^{4} e^{\operatorname {acot}{\relax (x )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} + \frac {24 x^{3} e^{\operatorname {acot}{\relax (x )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} - \frac {60 x^{2} e^{\operatorname {acot}{\relax (x )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} + \frac {44 x e^{\operatorname {acot}{\relax (x )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} - \frac {41 e^{\operatorname {acot}{\relax (x )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(acot(x))/(a*x**2+a)**3,x)

[Out]

-24*x**4*exp(acot(x))/(85*a**3*x**4 + 170*a**3*x**2 + 85*a**3) + 24*x**3*exp(acot(x))/(85*a**3*x**4 + 170*a**3
*x**2 + 85*a**3) - 60*x**2*exp(acot(x))/(85*a**3*x**4 + 170*a**3*x**2 + 85*a**3) + 44*x*exp(acot(x))/(85*a**3*
x**4 + 170*a**3*x**2 + 85*a**3) - 41*exp(acot(x))/(85*a**3*x**4 + 170*a**3*x**2 + 85*a**3)

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